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This semester I had a class about logic which was my favorite class. My college doesn't have any other classes that cover more advanced topics of logic, and as such I'm thinking about continuing to study logic on my own. In class, my teacher mentioned some more advanced things one can do with logic, like Model theory, applying topology to logic, etc...

These are the topics that we covered in class:

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Based on this, what topics should I learn next based on my background? In the long term, if the learning path isn't linear and branches off (like for example, after you learn general topology, you can begin to learn either Algebraic or Differential topology), what would that different specializations be?

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  • $\begingroup$ Since you're comfortable with first-order systems, consider looking into some weird non-classical logics, such as paraconsistent ones with more than two truth values. If you're okay with dabbling in physics, you might look into quantum logics, wherein the quantum measurement problem is solved by making the bold assertion that reality does not obey classical logic. $\endgroup$
    – Jerome
    Nov 10, 2022 at 19:56
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    $\begingroup$ logicmatters.net/tyl the study guide here is quite good. $\endgroup$
    – Ben Martin
    Nov 10, 2022 at 21:23
  • $\begingroup$ maybe have also a look at proof theory or several aspects of the philosophy of mathematics (consider consulting the stanford encyclopedia of philosophy for example on some branches like set theory, model theory...) $\endgroup$
    – Ettore
    Nov 11, 2022 at 16:46
  • $\begingroup$ You may have a look at John Stillwell's books. In particular What is mathematical logic? A good place to start : From Euclid to Godel. $\endgroup$ Nov 11, 2022 at 20:30

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Logic branches off in lots of directions, including:

  • Model theory: study of mathematical structures and the first-order logical formulas they satisfy.
  • Set theory: study of models of a particular axiom system (ZFC): often times set theory is concerned with the nebulously large infinite.
  • Recursion theory: if you add access to noncomputable functions to a programming language, this allows you to compute other noncomputable functions. This gives you a notion of one noncomputable function being "more powerful" than another. What is the structure of this relation?
  • Reverse mathematics: what axioms are actually necessary to prove what theorems? this can be answered by trying to prove the axioms from the theorems.

These each have very different feels to them. What was it that you liked about your logic class?

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